Strategic Network Creation for Enabling Greedy Routing

TL;DR

This paper presents the first game-theoretic network creation model that incorporates greedy routing, i.e., the strategic agents in the model are embedded in some metric space and strive for creating a network among themselves where all-pairs greedy routing is enabled.

Abstract

Today we rely on networks that are created and maintained by smart devices. For such networks, there is no governing central authority but instead the network structure is shaped by the decisions of selfish intelligent agents. A key property of such communication networks is that they should be easy to navigate for routing data. For this, a common approach is greedy routing, where every device simply routes data to a neighbor that is closer to the respective destination. Networks of intelligent agents can be analyzed via a game-theoretic approach and in the last decades many variants of network creation games have been proposed and analyzed. In this paper we present the first game-theoretic network creation model that incorporates greedy routing, i.e., the strategic agents in our model are embedded in some metric space and strive for creating a network among themselves where all-pairs greedy routing is enabled. Besides this, the agents optimize their connection quality within the created network by aiming for greedy routing paths with low stretch. For our model, we analyze the existence of (approximate)-equilibria and the computational hardness in different underlying metric spaces. E.g., we characterize the set of equilibria in 1-2-metrics and tree metrics and show that Nash equilibria always exist. For Euclidean space, the setting which is most relevant in practice, we prove that equilibria are not guaranteed to exist but that the well-known $Θ$-graph construction yields networks having a low stretch that are game-theoretically almost stable. For general metric spaces, we show that approximate equilibria exist where the approximation factor depends on the cost of maintaining any link.

Figures (2)

• Figure 1: Greedy paths in the Euclidean plane. Two such paths from $u$ to $v$ exist: $u,a,b,c,v$ and $u,y,z,v$. The latter shows, that even a single edge in a greedy path can be longer than $d(u,v)$. Path $u,w,v$ is not a greedy path, since $d(w, v) > d(u,v)$. The shortest greedy path is $u,a,b,c,v$, so we have $d^{\text{greedy}}_{G}(u,v) =2\sqrt{2}+4 \approx 6.83$. This yields $\text{stretch}_{G}(u, v) = d^{\text{greedy}}_{G}(u,v)/ d(u,v) =\frac{2\sqrt{2}+4}{6} \approx 1.14$. Note that for small enough $\varepsilon$ and $\delta$ the path $u,w,v$ has length less than $d^{\text{greedy}}_{G}(u,v)$. Thus, the shortest $u$-$v$-path in the network may not be a greedy path.
• Figure 2: Examples of DSGs ((b)-(e)) where only 1-edges and outgoing edges of node $a$ (colored orange) are shown, agent $a$ has stretch 1 to green colored nodes. (a) Sample 1-2-metric where only 1-edges are shown; (b) MinDSG for (a), 1-edges are colored black and 2-edges are colored orange; (c) the unique MaxDSG for (a); (d) BDSG(1) for (a); (e) some other DSG for (a); (f) a network that is not a DSG, since condition (iii) is violated: $|W_2^+(d)|$ can be decreased without increasing $|N(d)|$, by swapping the edge $(a,g)$ to $(a,m)$. Note that for node $a$ the set $\{b,c,d,e,f ,g\}$ is a minimum size dominating set for $G_{-a}^1$, but this set cannot be agent $a$'s strategy in a BDSG since $0.5|W_2(a)|+ 0.5|W_2^+(a)|$ is not minimal.

Theorems & Definitions (32)

• lemma 1
• Remark 1
• lemma 2
• theorem 2
• definition 1
• definition 2
• definition 3
• definition 4
• lemma 3
• lemma 4